On the automaticity of the Hankel determinants of a family of automatic sequences

TL;DR

This paper investigates the automaticity of Hankel determinants modulo 2 for certain automatic sequences, extending understanding of their algebraic and combinatorial properties.

Contribution

It establishes that the reduced Hankel determinants modulo 2 of specific automatic sequences are themselves automatic, revealing new structural insights.

Findings

Hankel determinants of these sequences are automatic modulo 2.

The results connect automatic sequences with their Hankel determinants.

Provides partial answers to the behavior of Hankel determinants of algebraic series.

Abstract

Hankel determinants and automatic sequences are two classical subjects widely studied in Mathematics and Theoretical Computer Science. However, these two topics were considered totally independently, until in 1998, when Allouche, Peyri\`ere, Wen and Wen proved that all the Hankel determinants of the Thue-Morse sequence are nonzero. This property allowed Bugeaud to prove that the irrationality exponents of the Thue-Morse-Mahler numbers are exactly 2. Since then, the Hankel determinants of several other automatic sequences, in particular, the paperfolding sequence, the Stern sequence, the period-doubling sequence, are studied by Coons, Vrbik, Guo, Wu, Wen, Bugeaud, Fu, Han, Fokkink, Kraaikamp, and Shallit. On the other hand, it is known that the Hankel determinants of a rational power series are ultimately zero, and the Hankel determinants of a quadratic power series over finite fields are ultimately periodic. It is therefore natural to ask if we can obtain similar results about the Hankel determinants of algebraic series. In the present paper, we provide a partial answer to this question by establishing the automaticity of the reduced Hankel determinants modulo $2$ of $\pm 1$-automatic sequences with kernel of cardinality at most $2$.